Quantifying the intrinsic mechanical flexibility of crystalline materials

The flexibility of a solid reflects its ability to accommodate reversible changes in size or shape. While the term is commonly used in describing physical and biological systems, a quantitative measure and hence the fundamental understanding of flexibility are presently lacking. Drawing on the phenomenology of flow in liquids, we introduce here a measure of intrinsic flexibility of crystalline materials as the fractional release of elastic stress or strain-energy through symmetry-constrained internal structural rearrangements. This metric distinguishes robustly the concept of flexibility from that of compliance. Using first-principles density functional theory calculations, we determine the flexibility of four key systems spanning a range of elastic stiffness and underlying chemistries. We find flexibility arises not only from large structural rearrangements associated with soft phonons, but also from hard phonons that couple strongly to strain fields. Our flexibility measure enables high-throughput screening of materials databases to identify next-generation ultra-flexible material.